
The distribution function of a random variable X distributed according to the Student's t distribution with n ≥ 1 degrees of freedom is a continuous function, F(x) = P(X < x), given by F(x) =  [ 1 / ( n^{½} B(1/2,n/2) ) ] ∫_{∞}^{x} (1 + t² / n) ^{ (n+1) / 2} dt 


where B() is the beta function.
The corresponding probability density function, f(x) = dF(x)/dx, is f(x) =  [ 1 / ( n^{½} B(1/2,n/2) ) ] (1 + x² / n) ^{ (n+1) / 2} 


For a large number of degrees of freedom, n >> 1, the distribution of [ X ( 1  1 / ( 4n ) ) ] / √[ 1 + X^{2} / ( 2n ) ] 

approaches that of a standard normal distribution. The problem of deciding how large n should be depends on the nature of the problem. The twosided Student's t distribution is given as A(x) = P( x < X < x ) = P( X < x ) 

where X has a Student's t distribution and x ≥ 0.
 double Student_t_Distribution( double x, int n )
This function returns F(x) where F(x) is described above.
 double Student_t_Density( double x, int n )
This function returns f(x) where f(x) is described above.
 void Student_t_Distribution_Tables( int n, double start, double delta,int nsteps, double *density, double* distribution_function )
This function returns f(x) where f(x) is described above in the array density, i.e. density[i] = f(x_{i}) where x_{i} = start + i delta, i = 0,...,nsteps and returns F(x) where F(x) is described above in the array distribution_function, i.e. distribution_function[i] = F(x_{i}) where x_{i} = start + i delta, i = 0,...,nsteps. Note that density must be declared double density[N] where N ≥ nsteps + 1 in the calling routine and similarly the distribution_function must be declared double distribution_function[N] where N ≥ nsteps + 1 in the calling routine.
 double Student_t_Distribution_Large_dof( double x, int n )
This function returns the asymptotic distribution of F(x) for a large number of degrees of freedom, n.
 double Absolute_Student_t_Distribution( double x, int n )
This function returns A(x) where A(x) is described above.
 double Absolute_Student_t_Distribution_Large_dof( double x, int n )
This function returns the asymptotic distribution of A(x) for a large number of degrees of freedom, n.
C source code is available for these routines:
 The file, student_t_distribution.c, contains the source code for the function Student_t_Distribution.
Dependencies: In addition to the the file student_t_distribution.c, the function Student_t_Distribution() requires the following files:
 The file, student_t_density.c, contains the source code for the function Student_t_Density.
Dependencies: In addition to the the file student_t_density.c, the function Student_t_Density() requires the following files:
 The file, student_t_dist_tables.c, contains the source code for the function Student_t_Distribution_Tables.
Dependencies: In addition to the the file student_t_dist_tables.c, the function Student_t_Distribution_Tables() requires the following files:
 The file student_t_distribution.c and its dependencies given above.
 The file student_t_density.c and its dependencies given above.
 The file, student_t_distribution_large_dof.c, contains the source code for the function Student_t_Distribution_Large_dof.
Dependencies: In addition to the file student_t_distribution_large_dof.c, the function Student_t_Distribution_Large_dof() requires the following file:
 The file, abs_student_t_distribution.c, contains the source code for the function Absolute_Student_t_Distribution
Dependencies: In addition to the the file abs_student_t_distribution.c, the function Absolute_Student_t_Distribution() requires the following files:
 The file, abs_student_t_distribution_large_dof.c, contains the source code for the function Absolute_Student_t_Distribution_Large_dof.
Dependencies: In addition to the file abs_student_t_distribution_large_dof.c, the function Absolute_Student_t_Distribution_Large_dof() requires the following file:
 The file, teststudenttbook.c, contains a test program of Student_t_Distribution( ), Student_t_Density(), and Absolute_Student_t_Distribution() written in C. This test program requires the files student_t_distribution.c, student_t_density.c, abs_student_t_distribution.c, and their dependencies listed above.
 The file, StudenttDistributionBook.txt, contains the results of the test program teststudenttbook.c.
 The file, teststudenttbook.sh, contains the shell script used to compile, link, and execute the test program teststudenttbook.c.
 The file, teststudenttdist.c, contains a test program of Student_t_Distribution( ) written in C. This test program requires the file student_t_distribution.c and its dependencies listed above.
 The file, StudenttDistributionTest.txt, contains the results of the test program teststudenttdist.c.
 The file, teststudenttdist.sh, contains the shell script used to compile, link, and execute the test program teststudenttdist.c.
 The file, teststudenttdensity.c, contains a test program of Student_t_Density( ) written in C. This test program requires the file student_t_density.c and its dependencies listed above.
 The file, StudenttDensityTest.txt, contains the results of the test program teststudenttdensity.c.
 The file, teststudenttdensity.sh, contains the shell script used to compile, link, and execute the test program teststudenttdensity.c.
 The file, teststudentttables.c, contains a test program of Student_t_Distribution_Tables( ) written in C. This test program requires the files student_t_distribution.c, student_t_density.c, and their dependencies listed above.
 The file, StudenttTablesTest.txt, contains the results of the test program teststudentttables.c.
 The file, teststudentttables.sh, contains the shell script used to compile, link, and execute the test program teststudentttables.c.
 The file, teststudenttasymptotic.c, contains a test program of Student_t_Distribution_Large_dof( ) written in C. This test program requires the files student_t_distribution_large_dof.c, student_t_distribution.c and their dependencies listed above.
 The file, StudenttDistributionAsymptoticTest.txt, contains the results of the test program teststudenttasymptotic.c.
 The file, teststudenttasymptotic.sh, contains the shell script used to compile, link, and execute the test program teststudenttasymptotic.c.
 The file, testabsstudenttasymptotic.c, contains a test program of Absolute_Student_t_Distribution_Large_dof( ) written in C. This test program requires the files abs_student_t_distribution_large_dof.c and its dependencies listed above.
 The file, AbsStudenttDistributionAsymptoticTest.txt, contains the results of the test program testabsstudenttasymptotic.c.
 The file, testabsstudenttasymptotic.sh, contains the shell script used to compile, link, and execute the test program testabsstudenttasymptotic.c.
